A proof of the quantum data processing inequality with a combinatorial flavour
Ashwin Nayak, Institute for Quantum Computing
The quantum data processing inequality (equivalently, the strong sub-additivity of von Neumann entropy) is a cornerstone of quantum information theory. It has been proven in numerous ways, each proof highlighting different aspects of the property.We present a proof of the data processing inequality based on elementary probability theory and properties of quantum states. In fact, the property follows from a strengthening of the substate theorem [Jain, Radhakrishnan, Sen'02] in the asymptotic setting. We prove, via an explicit construction, that in the limit of large n, the relative entropy of n copies each of two quantum states \rho, \sigma essentially equals their smooth max-relative entropy. This is analogous to the relationship between Shannon entropy and min-entropy that arises in the context of the noiseless coding theorem.
This is joint work with Shitikanth Kashyap and Michael Saks.